Nonlinear Dynamical Systems in
Re: Comment on Guillot
Thank you very much for your comment. Indeed your results on the locomotion of rats seem to corroborate mine on the behavioral sequences in mice, and I would greatly appreciate receiving a draft of your paper, if it is already available.
Considering that we postulated that the energy costs per sec were the same for identical acts, the dynamics of the lower level of control did not seem to be completely stochastic: It is due to the presence of « melodies » - deterministic sequences of acts - that have been previously evidenced by us, but also by others authors on laboratory and wild rodents (i.e. Kavanau, 1963 ; Baumgardner et al., 1980 ; Berridge et al., 1987).
Because of the shortness of each behavioral sequence (12 hours long, containing only 2 to 7 activity bouts), we have not been able to analyze with the same methods the dynamics of the higher level of control, which is responsible for the ultradian alternations of rest and activity bouts. A simple linear spectral analysis suggested their possible periodicity, but appropriate nonlinear methods might reveal that this system actually has a chaotic component of low dimensionality. We think that the dissipation of this system - compared to the previous one - may be due to the long-term learning of physical (temperature, luminosity), physiological (energy depletion, repletion) or ecological (social factors, predation) pacemakers. But what we interpreted as the dissipation of control systems may be due to the lack of granularity of the behavioral observations. In our case, the discrete nature of the behavioral records and of their translation into energy costs is a limitation for the exploration of such hypotheses.
Little is known about the implementation of the two levels of control of action selection in the brain. Hypotheses are formulated, for example, in terms of intrinsic Basic Rest Activity Cycle (Kleitman, 1980), or of multiple interacting oscillators (Barrio et al., 1997; Lloyd, 1997). A way to test such hypotheses is to search - by genetic programming, for example - what formal neural networks might be responsible of the organization of such behavioral sequences, to analyze their complexity, and to implement them on a real robot, situated in a real environment.
On Wed Dec 9, William Sulis wrote
>I read this paper with great interest having particpated, with Henry Szechtman and David Eilam in their work on locomotor behavior in rats treated with chronic quinpirole. In that study, to appear in Behavioral Neuroscience, we used ergodic theoretic methods to study the temporal organization of locomotor behavior in rats, allowed to move freely within an open field, following chronic injections of quinpirole, a dopamine agonist. Not included in that study were some suggestive results which bear some similarity to those being reported here. We found weak evidence for two levels of control governing the spatiotemporal organization of such locomotor activity. At the lowest level, responsible for the selection of individual actions, we found evidence of simple stochasticity, or at the very least, very high dimensional chaos. Hurst exponents were near 0.5, suggesting that the sequencing was identical to a random walk on the action space. The higher level concerned the global organization of activity appearing as the block size was increased. Markov entropies decreased indicating an increase in order and the Hurst exponent increased, again showing a departure from simple Brownian motion. However both remained high so that although structure was emerging, it was at best due to high dimensional chaos though of lower dimensionality than for the lower system. It suggested to us that stochasticity was an inherent feature of the action selection system and that what was critical were the sources of correlation and order appearing from these higher level systems. In your paper you provide evidence for an even higher level order emerging from circadian pacemakers.
>These sources of order appear in the temporal structure of the data. A intersting question concerns how they are effected at the neruonal or systemic level and whether they correspond to anatomical hierarchies, or whether these hierachies are purely functional.
Mon Dec 14